Explanation Given: v, w and u are vectors in R 3 The following quantities, a) v × ( u + w ) b) ( u + w ) ⋅ ( v × w ) c) ( u × w ) + ( w − v ) Formula used: Vector dot product The dot product of the given two vectors a = 〈 a i , a j , a k 〉 and b = 〈 b i , b j , b k 〉 can be calculated using the formula below, a ⋅ b = a i × b i + a j × a j + a k × b k Given the magnitude of the vectors ‖ a ‖ and ‖ b ‖ and the angle between the vectors θ , the dot product can be calculated using the following formula, a ⋅ b = ‖ a ‖ × ‖ b ‖ × cos θ The dot product of two vectors results in a scalar. Vector cross product The cross product of two vectors a and b , when their magnitudes ‖ a ‖ and ‖ b ‖ and the angle between them θ are known is given by the formula, a × b = ‖ a ‖ ⋅ ‖ b ‖ ⋅ sin ( θ ) n Where n is the unit vector at right angles to both a and b The cross product of the given two vectors a = 〈 a 1 , a 2 , a 3 〉 and b = 〈 b 1 , b 2 , b 3 〉 can also be calculated using the formula below, a × b = 〈 c 1 , − c 2 , c 3 〉 where c 1 = a 2 b 3 − a 3 b 2 c 2 = a 1 b 3 − a 3 b 1 c 3 = a 1 b 2 − a 2 b 1 Considering i, j and k to be the standard unit vectors in R 3 (assuming the right handed coordinate system), we can write the cross product in the form of the determinant as follows, a × b = | i j k a i a j a k b i b j b k | The cross product of two vectors is another vector that is at right angles to both. From the above discussion, we can conclude that the following The sum of two vectors is a vector Any scalar operation on a vector results in a vector Dot product of two vectors results in a scalar Cross product of two vectors results in a vector Calculation: Description a) v × ( u + w ) v × ( u + w ) According to property 1, the sum of two vectors is a vector...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 13, Problem 32CRE

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Which of the given quantities is a scalar

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Chapter 13, Problem 31CRE

Chapter 13, Problem 33CRE

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Chapter 13 Solutions

Calculus - Standalone book

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

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Which of the given quantities is a scalar

Solution Summary: The author explains the formula used to determine which of the given quantities is a scalar quantity.

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Which of the given quantities is a scalar

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Chapter 13 Solutions